# The Four Methods of Visualizing .9999... = 1

### (And A Proof)

To Continue Directly to the Mathmematical Proof, Click Here

## Method One: Fractional / Decimal Equivalance and Addition

## Method Two: Algebraic Reasoning

## Method Three: Logic and Conceptual Thinking

If it was not equal to one then there would be a number between it and 1. What number would that be?

It *is* intuitive to think: No matter how many 9's
you add, you'll never get all the way to 1.
But that's how it seems if you think about moving *toward* 1. What if
you think about moving *away* from 1?

That is, if you start at 1, and try to move away from 1 and toward
0.99999..., how far do you have to go to get to 0.99999... ? Any step
you try to take will be too far, so you can't really move at all -
which means that to move from 1 to 0.99999..., you have to stay at 1.
When you think of 0.999... as being 'a little below 1', it's because
in your mind, you've stopped expanding it; that is, instead of
0.999999...
you're *really* thinking of
0.999...999
which is not the same thing. You're absolutely right that 0.999...999
is a little below 1, but 0.999999... doesn't fall short of 1 *until*
you stop expanding it. But you never stop expanding it, so it never
falls short of 1.

Can you prove that .9999... does not equal 1?

## Method 4: Arithmatic

Subtraction

Division

## The Actual Proof

Infinate Geometric Series